2.3 Solving Equations Containing Fractions and Decimals


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1 2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions Operations with real numbers (Chapter ) Solve equations containing decimals Combining like terms (.9) The Distributive Property (.0) Solving linear equations (2. and 2.2) Finding the least common denominator (??) INTRODUCTION In Section 2. we solved equations that contained fractions. For example, To solve w 2 5 = 8 5, we add 2 5 to each side: To solve 5 = 5 8 y, we multiply each side by 8 5 : w 2 5 = = 5 8 y w = = y w + 0 = = y w = 224 = y y = 24 In some equations, though, it is easier and more efficient to clear any and all fractions, thereby making all of the constants and coefficients into integers. x For example, the equation = 2 x is easier to solve after its fractions are cleared and it has only integer constants and coefficients: 2x + 24 = 9x 8. What is it that allows us to transform x = 2 x into 2x + 24 = 9x 8? It is. our ability to find the least common denominator (LCD) for the three fractions (the LCD is 8); 2. our ability to multiply integers and fractions;. our ability to use the Distributive Property; and 4. our ability to apply the Multiplication Property of Equality: Solving Equations Containing Fractions and Decimals page 2. 
2 The Multiplication Property of Equality We may multiply any nonzero number, c, to each side of an equation. If a = b, then c a = c b, c 0 x Applying the Multiplication Property of Equality to an equation such as = 2 x requires that we first prepare the equation by grouping each side as one quantity, using parentheses: x = ( 2 x ) It is then that we can apply the Multiplication Property of Equality and multiply each side by 8: x = 8 ( 2 x ) 8 The solving of this equation will be completed later in this section. To learn the process, let s start with some simpler equations. EQUATIONS CONTAINING FRACTIONS Let us start with an equation that contains just one fraction, 2x = 4 x + 9. It is possible to solve this equation by first adding  4 x to each side, but to avoid the timeconsuming work involved with fractions, it is often helpful to first clear the fraction or clear the denominator and work only with integers. Caution: Clearing the fractions requires us to multiply each full side of the equation each and every term by the same value, the common denominator. We do not multiply only the terms containing fractions. ~Instructor Insight To this point, students have never been asked to create their own parentheses, so this extra step of preparation is introduced. After Example, it will become part of the multiplication step. As stated above, and as shown in Example, we must prepare the equation for multiplication by grouping each side using parentheses: (2x ) = ( 4 x + 9 ) Solving Equations Containing Fractions and Decimals page
3 Example : Solve this equation by first clearing the fraction(s). 2x = 4 x + 9 Procedure: There is only one fraction. Multiply each side by 4 to clear the fraction. Answer: 2x = 4 x + 9 The LCD is 4. Prepare the equation by placing parentheses around each side. (2x ) = ( 4 x + 9 ) Multiply each side by 4. 4(2x ) = 4( 4 x + 9 ) Distribute 4, on each side, to each term. To multiply 4 to the fraction 4 x, it is helpful to write 4 as x 4 = 4 4 x x simplifies to x. 8x 4 = x + 6 Reduce this to standard form by adding x to each side. Verify the solution, 8: 8x + (x) 4 = x + (x) + 6 Simplify each side. 2x = 5x 4 = 6 Isolate the variable term by? adding +4 to each side. 2(8) = 5x = Simplify each side. 6 5x = 40 Divide each side by 5. 5? =? = 4 x (8) x 5 = 40 5 Simplify. 5? = x =8 5 = 5 Note: The two steps of. preparing the equation for multiplication by placing parentheses around each side, and 2. showing the multiplication by the LCD can be combined into one step, just as they are in the next example. Solving Equations Containing Fractions and Decimals page 2. 
4 If an equation contains more than one fraction, then to clear all fractions, we must multiply by the least common denominator (LCD) of all the denominators. If the fractions already have a common denominator, then we multiply each side by that common denominator, as shown in Example 2. Example 2: Solve this equation by first clearing the fractions. w 2 + = w Procedure: Answer: There is only one denominator, 2. Multiply each side by 2 to clear the fractions. w 2 + = w The LCD is 2. Prepare the equation by placing parentheses around each side. Multiply each side by 2. 2 w 2 + = 2 w Distribute 2, on each side, to each term. Write 2 as 2 when multiplying the fractions 2 w = 2 w w 2 simplifies to w; simplifies to 9. w + 2 = 2w + 9 Reduce this to standard form by adding 2w to each side. Verify the solution, 7: w + (2w) + 2 = 2w + (2w) + 9 w + 2 = 9 w (2) = 9 + (2) w = 7 Simplify each side. Isolate the variable term by adding 2 to each side. Simplify. w 2 + = w (7) 2 +? = ? = = 2 2 Solving Equations Containing Fractions and Decimals page
5 YTI a) Solve each equation by first clearing the fractions. Verify the solution. Use Examples and 2 as guides. x = x + 4 b) m = 4 5 m 2 c) x 5 4 = 2 2x 5 d) 2 + w = 8 w 2 FRACTIONS WITH DIFFERENT DENOMINATORS If the denominators are different, we must identify the LCD before we multiply. Then, to clear the fractions, we must multiply each side by the LCD. Solving Equations Containing Fractions and Decimals page
6 Example : Solve each equation by first clearing the fractions. a) x + = 5x 6 b) y = y 6 Procedure: First identify the LCD, then multiply each side of the equation by the LCD to clear the fractions. Answer: a) x + = 5x 6 The LCD is 6. Prepare the equation by placing parentheses around each side. Multiply each side by 6. 6 x + = 6 5x 6 Distribute 6, or 6, on each side, to each term. 6 x + 6 = 6 5x 6 6 Simplify. 2x + 6 = 5x 8 2x + (2x) + 6 = 5x + (2x) 8 6 = x 8 Reduce this to standard form by adding 2x to each side. Simplify. Isolate the variable term by adding 8 to each side = x Simplify. You finish it: Verify that 8 is the solution. 24 = x Divide each side by. 24 = x 8 = x x = 8 Solving Equations Containing Fractions and Decimals page
7 b) y = y The LCD is 2. Prepare the equation by placing 6 parentheses around each side. Multiply each side by 2. 2 y = 2 y 6 Distribute 2, or 2, on each side, to each term. 2 y = 2 y 2 6 Simplify. y + = 4y 2 Reduce this to standard form by adding y to each side. y + (y) + = 4y + (y) 2 Verify the solution, : = y 2 Isolate the variable term by adding +2 to each side. y 4 +? = 2 y = y Simplify. = y y = 4 +? = ? = 2 0 2? = = 5 6 YTI 2 a) Solve each equation by first identifying the LCD and clearing the fractions. Verify the solution. Use Example as a guide. y 4 6 = y b) p p 6 = p + 2 Solving Equations Containing Fractions and Decimals page
8 c) x = x 4 5 d) w = 2 w 6 EQUATIONS CONTAINING DECIMALS Recall from Section.2 that terminating decimals are rational numbers (fractions) in which the denominators are powers of 0, such as 0, 00, and so on. For example, 0. = 25 0 and 0.25 = 00. Consider an equation that contains these two fraction: 0 x = x +. We can clear the fractions by multiplying each side by the LCD of 00, changing it to an equation of integers: 0x = 25x If this same equation is written with decimals instead of fractions, it would be 0.x = 0.25x +. Because this is the same equation, we also can multiply each side by 00, but this time we will clear the decimals. One major distinction, when clearing decimals, is to prepare the equation by first writing each constant and coefficient with the same number of decimal places. For example, each number in the equation 0.x = 0.25x + can be written with two decimal places: For 0., we can place one zero at the end of the number: 0. = 0.0 For, we can place a decimal point and two zeros at the end of the number: = already has two decimal places, so no change is necessary. Solving Equations Containing Fractions and Decimals page
9 The equation becomes 0.0x = 0.25x Now having two decimal places, each number is in terms of hundredths, and we can clear the decimals by multiplying each side by 00: 00(0.0x) = 00(0.25x +.00) Multiplying by 00 has the effect of moving the decimal point two places to the right. 0x = 25x + 00 It is now an equation of integers, and we can solve it using the techniques learned earlier in this chapter. Preparing an equation by creating an equal number of decimal places is an important first step when clearing decimals in an equation. Example 4: For each equation, What number of decimal places should each constant and coefficient have? Prepare the equation by building up each number, as necessary. By what number should we multiply each side of the equation to clear the decimals? a) 0.4x.2 = 0.5x b) 0.2y = 0.095y 0.9 Procedure: For each equation, the constant or coefficient with the highest number of decimal places indicates the number of decimal places each should have. a) 0.5 has two decimal places so we should build up each number to have two decimal places b) has three decimal places so we should build up each number to have three decimal places Number of Multiply decimal places New equation each side by Answer: a) Two 0.40x.20 = 0.5x b) Three 0.20y.000 = 0.095y 0.900,000 Solving Equations Containing Fractions and Decimals page
10 YTI For each equation, Decide the number of decimal places each constant and coefficient should have; Prepare the equation by building up each number, as necessary; and Decide what number to multiply each side of the equation to clear the decimals. Number of Multiply decimal places New equation each side by a) 2w 0.4 = +.8w b) 0.7k 0.4 = 0.25k c) 0.27v.6 = 0.2v 2 d) 0.x = 0.08x Example 5: Solve the equation by first clearing the decimals. a) 0.4x.2 = 0.5x b) 0.2y = 0.095y 0.9 Procedure: Use the information from the pervious example to prepare the equation for clearing the decimals. Answer: a) 0.4x.2 = 0.5x x.20 = 0.5x (0.40x.20) = 00(0.5x ) 40x 20 = 5x x + (5x) 20 = 5x + (5x) + 80 Write each decimal so that it has two decimal places. Prepare the equation by placing parentheses around each side. Multiply each side by 00. Distribute. Multiplying by 00 will clear all decimals. Reduce this to standard form by adding 5x to each side. Simplify. Solving Equations Containing Fractions and Decimals page
11 25x 20 = 80 Isolate the variable term by adding 20 to each side. 25x = Simplify. Verify the solution, 8: 25x = 200 Divide each side by (8).2? = 0.5(8) x 25 = = 8.2.2? = x = = 2.0 b) 0.2y = 0.095y y.000 = 0.095y 0.900,000(0.20y.000) =,000(0.095y 0.900) 20y,000 = 95y y + (95y),000 = 95y + (95y) y,000 = 900 Write each decimal so that it has three decimal places. Prepare the equation by placing parentheses around each side. Multiply each side by,000. Distribute. Multiplying by,000 will clear all decimals. Reduce this to standard form by adding 95y to each side. Simplify. Isolate the variable term by adding,000 to each side. 25y,000 +,000 = ,000 Simplify. You finish it: Verify that 4 is the solution. 25y = 00 Divide each side by y 25 = = 4 x = 4 Solving Equations Containing Fractions and Decimals page 2. 
12 YTI 4 Solve the equation by first clearing the decimals. Verify the solution. Use Example 5 as a guide. b) 2w 0.4 = +.8w b) 0.7k 0.4 = 0.25k c) 0.27v.6 = 0.2v 2 d) 0.x = 0.08x SOLVING EQUATIONS: THE ULTIMATE GUIDELINES Here is a summary of the steps involved in solving a variety of linear equations. Not all steps will be necessary for each equation; you should apply the guidelines in the order presented here but may skip any guideline that does not apply. For example, if an equation has no fractions, you may skip guideline (2) and proceed to guideline (). The Preparation: Solving Linear Equations: The Ultimate Guidelines. Eliminate any parentheses by distributing. 2. Clear any fractions or decimals by multiplying each side by the equation s LCD.. Combine like terms on each individual side. Isolating the Variable: 4. If necessary, reduce the equation to standard form. 5. If necessary, isolate the variable term then finish solving. Solving Equations Containing Fractions and Decimals page
13 The Ultimate Guidelines say that parentheses should be cleared first. This is true even if fractions are involved; in other words, if an equation has both fractions (or decimals) and parentheses, then it is best to clear the parentheses before trying to clear any fractions (or decimals). Example 6: Solve each equation and verify the solution. a) 2 x + 2 = (x ) b) 0.2(y 5) = 0.5(2y + ) 0.85 Procedure: First distribute, then clear the fractions or decimals. Answer: a) 2 x + 2 = (x ) Distribute and simplify; 2 2 = 2 x + = x The LCD is 6. Prepare the equation by placing parentheses around each side. Multiply each side by x + = 6(x ) Simplify: 6 2 x = x and 6 = 2. x + 2 = 8x 8 x + (x) + 2 = 8x + (x) 8 2 = 5x 8 Reduce this to standard form by adding x to each side. Simplify. Isolate the variable term by adding +8 to each side = 5x Verify the solution 4 : 20 = 5x 20 5 = 5x Divide each side by 5. 5 Simplify. 4 = x x = 2 ( x + 2 ) = (x ) 4 2 ( + 2 ) = 4 ( ) 2 ( 6 ) = 4 ( ) ( ) = = Solving Equations Containing Fractions and Decimals page 2. 
14 b) 0.2(y 5) = 0.5(2y + ) 0.85 Distribute. 0.6y.0 = 0.0y y.00 = 0.0y Write each decimal so that it has two decimal places. Prepare the equation by placing parentheses around each side. Multiply each side by (0.60y.00) = (0.0y )00 60y 00 = 0y y 00 = 0y 40 Combine like terms on the right side. Reduce this to standard form by adding 0y to each side. 60y + (0y) 00 = 0y + (0y) 40 Simplify. You finish it: Verify that 2 is the solution. 0y 00 = 40 Add 00 to each side. 0y = y = 60 Add 00 to each side. Divide each side by 0. 0y 0 = 60 0 y = 2 Solving Equations Containing Fractions and Decimals page
15 YTI 5 Solve each equation and verify the solution. Use example 6 as a guide. a) 2 (2h ) = 2h + 2 b) 0.5(p + ) = ( p) c) 8 (y + 2) = 4 2y d) 0.6(0n ) =.5(n + 2) 0. Solving Equations Containing Fractions and Decimals page
16 Answers: You Try It and Think About It YTI : a) x = 6 b) m = 5 c) x = 0 d) w = YTI 2: a) y = 6 b) p = 4 c) x = d) w =  YTI : a) One; 2.0w 0.4 =.0 +.8w; 0 b) Two; 0.7k 0.4 = 0.25k ; 00 c) Two; 0.27v.60 = 0.2v 2.00; 00 d) Three; 0.00x = 0.080x + 0.4;,000 YTI 4: a) w = 7 b) k = 6 c) v = 8 d) x = 7 YTI 5: a) h = 2 b) p = 60 c) y =  d) n = Think About It: There are no Think About It exercises in this section. Section 2. Exercises Think Again.. Consider the equation 2x + = 4 2 x + 4. What is the least common denominator on the right side of this equation? 2. If an equation contains decimals, why is it helpful for all of the constants and coefficients to have the same number of decimal places? Focus Exercises. Solve each equation. Verify your answer.. x + x 4 = 7 4. t 6 = 2 t 5. y y 4 = x = 5 x Solving Equations Containing Fractions and Decimals page
17 7. z + 5 = z 5 8. y 6 = y h = 4 h 2w w + 5 = 5 9. w + 7 = 6w x 5 = 4x 5. y = 6 5y 8 4. m = 4m n 8 = 4n w + 5 = 8w y 2 9 = 2y p p 8 = p x = 2 2 x 20. y + 4 = y r + 9 = 2r 2r z = z x = x t 9 4 = 2 + 5t 8 x + 5 x = 9 0 x p p = 2 p x.2 = x x 0.25 = 0.4x x = 0.7x x +.8 = 0.24x y 0. = y x =.2x 2.7 Solving Equations Containing Fractions and Decimals page
18 w = w r.25 = r v = v x = 0.x p 0.4 = p n 0. = n x.9 = 0.54x m = 0.06m x.2 = 0.8x v + 0.8v = 0.25v (5x ) = 2 x (2y + ) = y 6 ( 6x) =  6x t = 4 5 t (m + 8) 6 = 2 m (x + 5) = 5( x) (p + ) = 0.5p y = 0.5(2y ) 0.6 Think Outside the Box: Solve each. 5. 2x 8 4 = x x = x x + 8 x 2 = x 5 6 = x 4 Solving Equations Containing Fractions and Decimals page
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